The logarithm is defined as the inverse function of the exponential. Thus, \(log_a b =c \Rightarrow a^c=b\). Since the right hand side will not define b for all values of c when a is negative, it is common to restrict a to the positive numbers. It is also common to not let a=1, for then b must equal one. However, the definition of a logarithm can be extended to negative a if we allow b to be complex. In this case, new and unusual rules and situations arise. In fact, I see nothing wrong with letting a be complex too. This is a perfect example of a function that can be extended, but is commonly restricted for more practical use.

Yes, for that particular value of c, but not for all values of c. And at this point I am still assuming that a, b, and c are real numbers. So while \((-2)^2\) is a real number, \((-2)^{(\frac{1}{2})}\) is not a real number.